Last week, we explored spatial dependency and methods for measuring it. This week, we will examine how spatial autocorrelation can impact our analyses, especially in models where independence assumptions are crucial. To address these dependencies, we will use the GWmodel library to calculate local summary statistics for geographical areas and to fit basic spatially explicit models to our data.
You can download the slides of this week’s lecture here: [Link].
This week we will investigate the political geography of England and Wales, focusing on the results of the July 2024 General Election, which was won by the Labour Party led by Keir Starmer. You will work with data extracted from two data sources: the constituency results from the election and socio-demographic information relating to age groups, economic status, and ethnic background from the 2021 Census, extracted using the Custom Dataset Tool. These datasets have been prepared and merged. Along with this dataset, we also have access to a GeoPackage that contains the boundaries of the parliamentary constituencies.
You can download both files below and save them in your project folder under data/attributes and data/spatial, respectively.
| File | Type | Link |
|---|---|---|
| England and Wales Parliamentary Constituencies GE2024 | csv |
Download |
| England and Wales Parliamentary Constituencies Boundaries | GeoPackage |
Download |
Open a new script within your GEOG0030 project and save this as w05-election-analysis.r.
Begin by loading the necessary libraries:
R code
# load libraries
library(tidyverse)
library(sf)
library(tmap)
library(GWmodel)
library(easystats)
library(spdep)You may have to install some of these libraries if you have not used these before.
Once downloaded, we can load both files into memory:
R code
# load election dataset
elec_24 <- read_csv("data/attributes/England-Wales-GE2024-Constituency-Vars.csv")Rows: 575 Columns: 28
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (6): constituency_code, constituency_name, region_name, winning_party, ...
dbl (22): eligible_voters, valid_votes, conservative_votes, labour_votes, li...
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# load constituency boundaries
cons_24 <- st_read("data/spatial/England-Wales-GE2024-Boundaries.gpkg")Reading layer `England-Wales-GE2024-Boundaries' from data source
`/Users/justinvandijk/Library/CloudStorage/Dropbox/UCL/Web/jtvandijk.github.io/GEOG0030/data/spatial/England-Wales-GE2024-Boundaries.gpkg'
using driver `GPKG'
Simple feature collection with 575 features and 8 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 85713.71 ymin: 7054.1 xmax: 655644.8 ymax: 657536.5
Projected CRS: OSGB36 / British National Grid# inspect
head(elec_24)# A tibble: 6 × 28
constituency_code constituency_name region_name winning_party eligible_voters
<chr> <chr> <chr> <chr> <dbl>
1 W07000081 Aberafan Maesteg Wales Lab 72580
2 E14001063 Aldershot South East Lab 78553
3 E14001064 Aldridge-Brownhil… West Midla… Con 70268
4 E14001065 Altrincham and Sa… North West Lab 74025
5 W07000082 Alyn and Deeside Wales Lab 75790
6 E14001066 Amber Valley East Midla… Lab 71546
# ℹ 23 more variables: valid_votes <dbl>, conservative_votes <dbl>,
# labour_votes <dbl>, libdem_votes <dbl>, conservative_vote_share <dbl>,
# labour_vote_share <dbl>, libdem_vote_share <dbl>,
# aged_15_years_and_under <dbl>, aged_16_to_24_years <dbl>,
# aged_25_to_34_years <dbl>, aged_35_to_49_years <dbl>,
# aged_50_years_and_over <dbl>, eco_not_applicable <dbl>,
# eco_active_employed <dbl>, eco_active_unemployed <dbl>, …# inspect
head(cons_24)Simple feature collection with 6 features and 8 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 368283 ymin: 100922.9 xmax: 524969.7 ymax: 393552.1
Projected CRS: OSGB36 / British National Grid
pcon24cd pcon24nm pcon24nmw bng_e bng_n long lat
1 E14001063 Aldershot 484716 155270 -0.78648 51.29027
2 E14001064 Aldridge-Brownhills 404720 301030 -1.93172 52.60704
3 E14001065 Altrincham and Sale West 374132 389051 -2.39049 53.39766
4 E14001066 Amber Valley 440478 349674 -1.39771 53.04282
5 E14001067 Arundel and South Downs 497309 118530 -0.61584 50.95798
6 E14001068 Ashfield 450035 356564 -1.25410 53.10395
globalid geom
1 {3F742659-208E-4859-8905-1CE5E2B87AAC} MULTIPOLYGON (((485406.9 15...
2 {4668E55F-5248-45E4-9927-30562A6812D5} MULTIPOLYGON (((406519.1 30...
3 {56FDBCDF-25DA-4EE5-AB66-6FC9C71D2927} MULTIPOLYGON (((379104.1 39...
4 {9F9742E9-AD8C-4878-9841-1DF40BBA2619} MULTIPOLYGON (((444868.4 35...
5 {4AFD2380-5BB7-4A01-8712-8637117E96D4} MULTIPOLYGON (((523813.2 11...
6 {E0D1319E-FD1E-43AF-B809-79127FB22B22} MULTIPOLYGON (((455809 3579...You can inspect both objects using the View() function.
In GEOG0018 Research Methods in Human Geography, we already worked with this electoral dataset and we hypothesised that predominantly older voters tend to support the Conservative Party. To explore this, we examined the relationship between the proportion of individuals over 50 years old in each parliamentary constituency (aged_50_years_and_over) and the proportion of votes cast for the Conservative Party (conservative_vote_share). A scatterplot and Pearson’s correlation revealed a moderate association between the two variables, suggesting a possible link between age demographics and Conservative voting patterns:
R code
# correlation
cor.test(elec_24$aged_50_years_and_over, elec_24$conservative_vote_share, method = "pearson")
Pearson's product-moment correlation
data: elec_24$aged_50_years_and_over and elec_24$conservative_vote_share
t = 16.472, df = 573, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.5086971 0.6199173
sample estimates:
cor
0.5668849 # scatterplot
plot(elec_24$aged_50_years_and_over, elec_24$conservative_vote_share, xlab = "Proportion of population over 50 years old",
ylab = "Proportion of votes for the Conservative party")
However, as you should know by now, the first step when working with spatial data is to map your variables:
R code
# join attribute data onto spatial data
cons_24 <- cons_24 |>
left_join(elec_24, by = c("pcon24cd" = "constituency_code"))
# shape, polygons
tm_shape(cons_24) +
# specify column, colours
tm_fill(
col = "conservative_vote_share",
style = "jenks",
n = "5",
palette = c("#f1eef6", "#bdc9e1", "#74a9cf", "#2b8cbe", "#045a8d"),
title = "Conservative vote share"
) +
# set layout
tm_layout(
legend.outside = FALSE,
legend.position = c("left", "top"),
frame = FALSE
)
R code
# shape, polygons
tm_shape(cons_24) +
# specify column, colours
tm_fill(
col = "aged_50_years_and_over",
style = "jenks",
n = "5",
palette = c("#f0f0f0", "#d9d9d9", "#bdbdbd", "#969696", "#737373"),
title = "50+ population share"
) +
# set layout
tm_layout(
legend.outside = FALSE,
legend.position = c("left", "top"),
frame = FALSE
)
Looking at the maps, the spatial distribution of Conservative vote share and the proportion of the population over 50 appears neither random nor uniform. This raises the question to what extent the observed correlation in Figure 1 holds consistently across different constituencies? To answer this we can run a Geographically Weighted Correlation using the GWmodel library.
Geographically Weighted Correlation (GWC) allows us to investigate whether the strength and direction of the association between variables vary across space. By applying a localised correlation technique, GWC calculates the correlation coefficient within specified spatial windows or kernels across the study area. This can reveal geographic areas where the relationship is stronger or weaker, providing more insight than a single global correlation and helping us to understand spatial heterogeneity in the data.
The GWmodel library allows us to calculate local statistics, such as means, standard deviations, variances, and correlations. But as with spatial autocorrelation, we must define what local means. One approach is to use a kernel function to select which values contribute to each local estimate. Kernels operate on point locations (e.g. polygon centroids) and apply a window with a specified shape and bandwidth. The bandwidth, which defines the kernel’s size, can be set in absolute terms (e.g. within 10 km) or in relative terms (e.g. the 10 nearest centroids), the latter known as an adaptive kernel.
The GWmodel library uses the older sp data format for handling spatial data. We therefore need to convert our current sf object to sp before continuing:
R code
# to sp
cons_24_sp <- as_Spatial(cons_24)The sf package is now preferred over sp in R for its modern, efficient handling of spatial data. By using simple features, a standardised format for spatial geometries, sf is more compatible with other geospatial tools and integrates smoothly with the tidyverse, simplifying data manipulation and analysis. However, some libraries still rely on sp and have not yet transitioned to sf.
We can now calculate a geographically weighted correlation. We will use an adaptive bandwidth that estimates the local correlation using the 25 nearest constituencies:
R code
# geographically weighted correlation
cons_24_cor <- gwss(cons_24_sp, vars = c("conservative_vote_share", "aged_50_years_and_over"),
bw = 25, adaptive = TRUE)We can now extract the values and bind these back to our original sf object:
R code
# extract correlation
cons_24 <- cons_24 |>
mutate(cons_age_cor = cons_24_cor$SDF$Corr_conservative_vote_share.aged_50_years_and_over)
# inspect
summary(cons_24$cons_age_cor) Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.02121 0.57869 0.72341 0.68576 0.83984 0.95214 The results of the outcomes of gwss function can be accessed through the $SDF data frame.
The summary shows that there indeed seems to be variation of this relationship across the country, with the local Pearson correlation ranging from very weak to very strong. Of course, we can map this in the usual way:
R code
# shape, polygons
tm_shape(cons_24) +
# specify column, colours
tm_fill(
col = "cons_age_cor",
style = "jenks",
n = "5",
palette = c("#fee5d9", "#fcae91", "#fb6a4a", "#de2d26", "#a50f15"),
title = "Local correlation"
) +
# set layout
tm_layout(
legend.outside = FALSE,
legend.position = c("left", "top"),
frame = FALSE
)
conservative_vote_share and aged_50_years_and_over variables.While the map shows correlation patterns, it doesn not indicate statistical significance. To evaluate this, we can use a Monte Carlo simulation with the gwss.montecarlo() function. However, for a large dataset like ours, this process is computationally intensive and time-consuming, especially with more than the default 99 simulations.
Although we will not run a Monte Carlo simulation here, the code below demonstrates how to run this and link the results back to the cons_24 spatial dataframe
R code
# geographically weighted correlation, monte carlo simulation
cons_24_cor_sig <- gwss.montecarlo(cons_24_sp, vars = c("conservative_vote_share",
"aged_50_years_and_over"), bw = 25, adaptive = TRUE, nsim = 99) |>
as_tibble() |>
select(Corr_conservative_vote_share.aged_50_years_and_over)
# replace names
names(cons_24_cor_sig) <- "cons_age_cor_p"
# bind results
cons_24 <- cons_24 |>
cbind(cons_24_cor_sig) |>
mutate(cons_24_cor = if_else(cons_age_cor_p < 0.025, cons_age_cor_p, if_else(cons_age_cor_p >
0.975, cons_age_cor_p, NA)))It took more than 2 minutes to run the Monte Carlo simulation on an Apple MacBook Pro M1 (16GB RAM) with the default of 99 simulations.
A correlation describes the strength and direction of a linear relationship, but if we want to quantify the change in a dependent variable (y) for a one-unit change in the independent variable(s) (x) we need to run a regression. Geographically weighted regression (GWR) is used when the relationship between a dependent and set of independent variables is not constant across space, meaning the model coefficients vary by location. This is useful when you suspect that the relationship between variables may change depending on the geographic context. GWR provides a localised understanding of the relationships by allowing each observation to have its own set of regression coefficients, which can provide insights into how relationships differ across the study area.
GWR fits a separate regression equation at each location in the study area, weighting nearby observations more heavily than those farther away. Again, the weighting is typically based on a kernel function. The basic GWR equation is:
\[ y_{i} = \beta_{0}(\upsilon_{i}, v_{i}) + \sum_{k=1}^{p}\beta_{k}(\upsilon_{i}, v_{i})x_{ik} + \epsilon_{i} \]
where \((\upsilon_{i}, v_{i})\) are the coordinates of location \(i\) and \(\beta_{k}(\upsilon_{i}, v_{i})\) are the location-specific coefficients.
To predict the Conservative voter share, we hypothesise that certain socio-demographic variables may play a significant role. These independent variables include:
| Variable | Description |
|---|---|
eco_active_employed |
Proportion of economically active employed individuals. |
eco_active_unemployed |
Proportion of economically active unemployed individuals. |
eth_white |
Proportion of the population identifying as white. |
eth_black |
Proportion of the population identifying as black. |
aged_25_to_34_years |
Proportion of the population between ages 25 and 34. |
aged_50_years_and_over |
Proportion of the population over age 50. |
We can run the multivariate regression as follows:
R code
# regression
lm_voteshare <- lm(conservative_vote_share ~ eco_active_employed + eco_active_unemployed +
eth_white + eth_black + aged_25_to_34_years + aged_50_years_and_over, data = cons_24)
# summary
summary(lm_voteshare)
Call:
lm(formula = conservative_vote_share ~ eco_active_employed +
eco_active_unemployed + eth_white + eth_black + aged_25_to_34_years +
aged_50_years_and_over, data = cons_24)
Residuals:
Min 1Q Median 3Q Max
-0.286730 -0.040843 0.001182 0.044496 0.220419
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.16210 0.08779 -1.846 0.06535 .
eco_active_employed 1.36140 0.13208 10.308 < 2e-16 ***
eco_active_unemployed -0.32339 0.91429 -0.354 0.72369
eth_white -0.28028 0.03567 -7.857 1.97e-14 ***
eth_black -0.29751 0.10190 -2.920 0.00364 **
aged_25_to_34_years -1.52695 0.20954 -7.287 1.07e-12 ***
aged_50_years_and_over 0.59711 0.09827 6.076 2.26e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.06797 on 568 degrees of freedom
Multiple R-squared: 0.5537, Adjusted R-squared: 0.5489
F-statistic: 117.4 on 6 and 568 DF, p-value: < 2.2e-16The model summary indicates that most variables are statistically significant predictors of Conservative vote share. The results suggest that older and employed populations are linked to higher Conservative support, while younger age groups and certain ethnic proportions show a negative relationship.
We can now run the diagnostics to see whether the regression assumptions are met:
R code
# run diagnostics
performance::check_model(lm_voteshare, check = c("qq", "normality"))
R code
# run diagnostics
performance::check_model(lm_voteshare, check = c("linearity", "homogeneity"))
R code
# run diagnostics
performance::check_model(lm_voteshare, check = c("vif", "outliers"))
The model diagnostics seem generally fine, perhaps except for the linearity check. This could indicate a curved relationship or, alternatively, suggest spatial dependence in the relationship. If spatial dependence is present, it may imply that the observations are not independent. We can examine this by assessing whether the model residuals display spatial dependence. Let us begin with a map:
R code
# join model residuals onto spatial data
cons_24 <- cons_24 |>
mutate(lm_residuals = lm_voteshare$residuals)
# shape, polygons
tm_shape(cons_24) +
# specify column, colours
tm_fill(
col = "lm_residuals",
style = "jenks",
n = "5",
palette = c("#ffffb2", "#fecc5c", "#fd8d3c", "#f03b20", "#f03b20"),
title = "Regression residuals"
) +
# set layout
tm_layout(
legend.outside = FALSE,
legend.position = c("left", "top"),
frame = FALSE
)
Looking at Figure 8 there appears to be some spatial structure in the model residuals. We can test this more formally using Moran’s I test to assess spatial autocorrelation:
R code
# create neighbour list
cons_24_nb <- poly2nb(cons_24, queen = TRUE)Warning in poly2nb(cons_24, queen = TRUE): some observations have no neighbours;
if this seems unexpected, try increasing the snap argument.Warning in poly2nb(cons_24, queen = TRUE): neighbour object has 3 sub-graphs;
if this sub-graph count seems unexpected, try increasing the snap argument.# create spatial weights matrix
cons_24_nb_weights <- cons_24_nb |>
nb2listw(style = "W", zero.policy = TRUE)
# moran's test
moran <- moran.mc(cons_24$lm_residuals, listw = cons_24_nb_weights, nsim = 999)
# results
moran
Monte-Carlo simulation of Moran I
data: cons_24$lm_residuals
weights: cons_24_nb_weights
number of simulations + 1: 1000
statistic = 0.4155, observed rank = 1000, p-value = 0.001
alternative hypothesis: greaterThe Moran’s I test indicates significant spatial autocorrelation in our model residuals, suggesting that the observations are not independent. As a result, we can proceed with a Geographically Weighted Regression to account for spatial variation in the relationship between the variables.
We will first estimate the ‘optimal’ bandwidth using an automated bandwidth selection procedure:
R code
# bandwidth selection
cons_24_bw <- bw.gwr(conservative_vote_share ~ eco_active_employed + eco_active_unemployed +
eth_white + eth_black + aged_25_to_34_years + aged_50_years_and_over, data = cons_24,
adaptive = TRUE)Adaptive bandwidth: 363 CV score: 2.208986
Adaptive bandwidth: 232 CV score: 2.105893
Adaptive bandwidth: 151 CV score: 1.973309
Adaptive bandwidth: 100 CV score: 1.940199
Adaptive bandwidth: 70 CV score: 2.008461
Adaptive bandwidth: 120 CV score: 1.945603
Adaptive bandwidth: 89 CV score: 1.942555
Adaptive bandwidth: 108 CV score: 1.938444
Adaptive bandwidth: 112 CV score: 1.942082
Adaptive bandwidth: 104 CV score: 1.938601
Adaptive bandwidth: 109 CV score: 1.940144
Adaptive bandwidth: 106 CV score: 1.939139
Adaptive bandwidth: 108 CV score: 1.938444 The model can then be fitted.
R code
# run gwr
cons_24_gwr <- gwr.basic(conservative_vote_share ~ eco_active_employed + eco_active_unemployed +
eth_white + eth_black + aged_25_to_34_years + aged_50_years_and_over, data = cons_24,
adaptive = TRUE, bw = cons_24_bw)
# inspect
cons_24_gwr ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2025-02-11 09:27:19.19713
Call:
gwr.basic(formula = conservative_vote_share ~ eco_active_employed +
eco_active_unemployed + eth_white + eth_black + aged_25_to_34_years +
aged_50_years_and_over, data = cons_24, bw = cons_24_bw,
adaptive = TRUE)
Dependent (y) variable: conservative_vote_share
Independent variables: eco_active_employed eco_active_unemployed eth_white eth_black aged_25_to_34_years aged_50_years_and_over
Number of data points: 575
***********************************************************************
* Results of Global Regression *
***********************************************************************
Call:
lm(formula = formula, data = data)
Residuals:
Min 1Q Median 3Q Max
-0.286730 -0.040843 0.001182 0.044496 0.220419
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.16210 0.08779 -1.846 0.06535 .
eco_active_employed 1.36140 0.13208 10.308 < 2e-16 ***
eco_active_unemployed -0.32339 0.91429 -0.354 0.72369
eth_white -0.28028 0.03567 -7.857 1.97e-14 ***
eth_black -0.29751 0.10190 -2.920 0.00364 **
aged_25_to_34_years -1.52695 0.20954 -7.287 1.07e-12 ***
aged_50_years_and_over 0.59711 0.09827 6.076 2.26e-09 ***
---Significance stars
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.06797 on 568 degrees of freedom
Multiple R-squared: 0.5537
Adjusted R-squared: 0.5489
F-statistic: 117.4 on 6 and 568 DF, p-value: < 2.2e-16
***Extra Diagnostic information
Residual sum of squares: 2.624389
Sigma(hat): 0.06767633
AIC: -1451.196
AICc: -1450.941
BIC: -1940.526
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: bisquare
Adaptive bandwidth: 108 (number of nearest neighbours)
Regression points: the same locations as observations are used.
Distance metric: Euclidean distance metric is used.
****************Summary of GWR coefficient estimates:******************
Min. 1st Qu. Median 3rd Qu. Max.
Intercept -1.089291 -0.441131 -0.160803 0.209568 1.1375
eco_active_employed -1.045077 0.151632 0.930912 1.498222 2.8325
eco_active_unemployed -10.747683 -4.837597 0.830042 3.163451 9.8531
eth_white -1.411777 -0.365168 -0.230988 -0.109584 0.1738
eth_black -3.276551 -0.308415 -0.121533 0.071703 1.1200
aged_25_to_34_years -3.659487 -1.575036 -1.288467 -0.342851 1.9350
aged_50_years_and_over -0.287576 0.503695 1.075497 1.383006 2.1875
************************Diagnostic information*************************
Number of data points: 575
Effective number of parameters (2trace(S) - trace(S'S)): 103.6449
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 471.3551
AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): -1622.807
AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): -1730.89
BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): -1880.236
Residual sum of squares: 1.444834
R-square value: 0.7542715
Adjusted R-square value: 0.7001241
***********************************************************************
Program stops at: 2025-02-11 09:27:19.294264 The global regression results are derived from the standard OLS model, representing a single regression for the entire study area without accounting for spatial variation. In contrast, the geographically weighted regression (GWR) results stem from 575 separate but spatially overlapping regression models, each fitted with geographic weighting to different spatial subsets of the data, allowing for local variation in the regression estimates. We can also see that the model fit (adjusted \(R^2\)) has increased to 70%.
All the local estimates are contained in the spatial data frame cons_24_gwr$SDF. You can inspect the results using the View() function.
We can now also map all the local \(R^2\) values to get an idea of how well the model fit varies spatially:
R code
# join local r squared values onto spatial data
cons_24 <- cons_24 |>
mutate(gwr_r2 = cons_24_gwr$SDF$Local_R2)
# shape, polygons
tm_shape(cons_24) +
# specify column, colours
tm_fill(
col = "gwr_r2",
style = "jenks",
n = "5",
palette = c("#fee5d9", "#fcae91", "#fb6a4a", "#de2d26", "#a50f15"),
title = "Local R-squared"
) +
# set layout
tm_layout(
legend.outside = FALSE,
legend.position = c("left", "top"),
frame = FALSE
)
In similar fashion, and with some data wrangling, we can map the individual local coefficients of our independent variables:
R code
# join local coefficient values onto spatial data, select, pivot
cons_24_coef <- cons_24 |>
mutate(
eco_active_employed_coef = cons_24_gwr$SDF$eco_active_employed,
eco_active_unemployed_coef = cons_24_gwr$SDF$eco_active_unemployed,
eth_white_coef = cons_24_gwr$SDF$eth_white,
eth_black_coef = cons_24_gwr$SDF$eth_black,
aged_25_to_34_years_coef = cons_24_gwr$SDF$aged_25_to_34_years,
aged_50_years_and_over_coef = cons_24_gwr$SDF$aged_50_years_and_over
) |>
select(40:45) |>
pivot_longer(cols = !geom, names_to = "variable", values_to = "coefficient")
# shape, polygons
tm_shape(cons_24_coef) +
# specify column, colours
tm_fill(
col = "coefficient",
style = "jenks",
n = 9,
palette = c("#e66101", "#fdb863", "#cccccc", "#b2abd2", "#5e3c99"),
title = "Local GWR coefficient"
) +
# facet
tm_facets(
by = "variable",
ncol = 2
) +
# set layout
tm_layout(
legend.show = TRUE,
legend.outside = FALSE,
frame = FALSE,
panel.labels = c("Age: 25-34", "Age: >50", "Employed", "Unemployed", "Ethnicity: Black", "Ethnicity: White"),
)Use "fisher" instead of "jenks" for larger data setsVariable(s) "coefficient" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
Mapping these local estimates reveals an interesting geography, where, for example, the proportion of the population over 50 years old is a better predictor for the Conservative vote share in regions like the South West, Greater London, and parts of the North West and Yorkshire and the Humber.
Not all of the local regression estimates may be statistically significant. We can use use the estimated \(t\)-values, as indicated by the _TV suffix in the SDF data frame of the GWR results, to filter out those are not. For example, to filter at 95% confidence, remove \(t\)-values outside the range of -1.96 to +1.96.
For a more comprehensive dive into Geographically Weighted Regression, the Spatial Modelling for Data Scientists course by Liverpool-based Professors Francisco Rowe and Dani Arribas-Bel provides an excellent introduction.
In this tutorial, we have looked at geographically weighted correlation and regression. However, whilst out of the scope of this module, there are many other approaches to account for space in statistical models. Examples of such models are the spatial error model and the spatial lag model.
The spatial error model is used when the error terms in a regression model exhibit spatial autocorrelation, meaning the error terms are not independent across space. This can happen due to omitted variables that have a spatial pattern or unmeasured factors that affect the dependent variable similarly across nearby locations.
The model adjusts for spatial autocorrelation by adding a spatially lagged error term (a weighted sum of the errors from neighbouring locations) to the regression equation:
\[ y = X\beta + \upsilon, \upsilon = \lambda W \upsilon + \epsilon \]
where \(X\beta\) represents the standard regression components, \(\lambda\) is a spatial autoregressive parameter, \(W\) is a spatial weights matrix, and \(\upsilon\) is a vector of spatially autocorrelated errors.
Spatial error models can be fitted using R’s spatialreg package.
The spatial lag model is appropriate when the dependent variable itself exhibits spatial dependence. This means the value of the dependent variable in one location depends on the values in neighbouring locations. This model is used to capture the spillover effects or diffusion processes, where the outcome in one area is influenced by outcomes in nearby areas (e.g. house prices, crime rates).
The model incorporates a spatially lagged dependent variable, which is the weighted sum of the dependent variable values in neighbouring locations, into the regression equation:
\[ y = \rho Wy + X\beta + \epsilon \]
where \(\rho\) is the spatial autoregressive coefficient, \(Wy\) represents the spatially lagged dependent variable, and \(X\beta\) represents the standard regression components.
Spatial lag models can be fitted using R’s spatialreg package.
The spatial error model adjusts for spatial autocorrelation in the error terms, whereas the spatial lag model adjusts for spatial dependence in the dependent variable itself. The spatial error model does not alter the interpretation of the coefficients of the independent variables, while the spatial lag model introduces a feedback loop where changes in one area can influence neighbouring areas.
Both the spatial error and spatial lag models assume that the relationships between variables are the same across the study area, with adjustments made only for spatial dependencies. GWR, on the other hand, allows the relationships themselves to vary across space. GWR is more flexible but also more complex and computationally intensive, providing local instead of global estimates of coefficients.
The Geographically Weighted Regression (GWR) reveals some varying patterns in the sign and directiion of the Conservative voter share, but it is likely that other factors are at play. One could hypothesise that another potential predictor of the Conservative vote share is the proportion of the population employed in a particular industry. Try to do the following:
data directory. The csv, extracted using the Custom Dataset Tool, contains the number of employed individuals as recorded in the 2021 Census by their Standard Industrial Classification (SIC) code.| File | Type | Link |
|---|---|---|
| England and Wales Parliamentary Constituencies Employed by Industry | csv |
Download |
This week, we explored decomposing global measures to account for non-stationarity in spatial variables. Geographically Weighted Regression (GWR) is particularly useful when the relationship between variables varies across space, as it allows for localised regression coefficients and helps identify spatial heterogeneity that might be overlooked in global models. This was a rather heavy one, time for some light reading?